What is group theory?

Question

Dad often talks about it in for his Maths degree but I don't know what he's talking about and it would be nice if I could contribue/support him in the conversation!

Answer 1

Group theory is a branch of the so called "abstract algebra." It attempts to study some properties common to particular structures. The structures are defined through an operation its members have. One example of such structure is the set of all integers. Our operation there is the regular addition. Which means that if we add two integers we still stay within integers. We have a member, zero, adding which doesn't change a thing.
For every positive integer we have a sort of converse to it, which we call a negative number. We can add any three numbers in any order, and still obtain the same results. Those are the properties of the most useful mathematical structure known as a group. The benefit of studying an abstract group is that we can ignore the name of the particular elements but concentrate on the general principles they posess. There are many advantages to studying an abstraction, it makes generalization easier. Because groups are so general, they are objects we meet everywhere in nature. Other sciences, such as physics, biology, chemistry and geology, can benefit from a greater understanding of group theory. There are other algebraic structures like rings, fields, vector spaces or modules, but groups are the most useful. P. S Aleksandrov has a good introduction to group theory for bright highschool students.
Hope that helps some.

Answer 2

GROUP THEORY: Group theory is that branch of mathematics concerned with the study of groups (as defined below). It has several applications in physics and chemistry. Groups are used throughout mathematics, often to capture the internal symmetry of other structures, in the form of automorphism groups. An internal symmetry of a structure is usually associated with an invariant property; the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group. RING THEORY: In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. A ring is a generalization of the integers. Other examples include the polynomials and the integers modulo n. The branch of abstract algebra which studies rings is called ring theory.

Answer 3

In mathematics, group theory is the study of "groups", which are defined as sets containing elements (such as numbers), some binary operator (i.e., something you can "do" to two of the elements that gives you back a third element, such as addition), and where the following four conditions are satisfied: 1) it's closed under the operation, 2) the associative property holds for the operation, 3) there's an element called the "identity" element such that using the operator with this and any other element gives you the element back, 4) every element has an inverse element such that using the operation on the two gives you back the identity element.

So for example, let's take the set of all positive numbers for its elements, and "*" for the operator (multiplication). #1 is true because if we multiply any two numbers in our group, we get another positive number back, which must be in our group. #2 is true because if you pick any numbers A and B in our group, A*B and B*A give you the same answer. For #3 we have an identity element, namely the number 1, because A*1=A for any number A in our set of positive numbers. And for #4, we can pick any positive number A and find another positive number 1/A, which when multiplied together gives us 1. So the set of all positive numbers along with multiplication gives us a "group".

The interesting thing about group theory is that you're not limited to numbers. If you define your "elements", your "operation", your "identity" element and your "inverse" correctly, there are all sorts of things you can define as groups. You can use it with matrixes, shapes, certain physics behavior, etc. All the laws of numbers and geometry are self-consistant, so it turns out you can apply these rules to things that aren't numbers.

Answer 4

Group theory is that branch of mathematics concerned with the study of groups. It has several applications in physics and chemistry.

Groups are used throughout mathematics, often to capture the internal symmetry of other structures, in the form of automorphism groups. An internal symmetry of a structure is usually associated with an invariant property; the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group.

In Galois theory, which is the historical origin of the group concept, one uses groups to describe the symmetries of the equations satisfied by the solutions to a polynomial equation. The solvable groups are so-named because of their prominent role in this theory.

Abelian groups underlie several other structures that are studied in abstract algebra, such as rings, fields, and modules.

In algebraic topology, groups are used to describe invariants of topological spaces (the name of the torsion subgroup of an infinite group shows the legacy of this field of endeavor). They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. Examples include the fundamental group, homology groups and cohomology groups.

The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they combine analysis and group theory and are therefore the proper objects for describing symmetries of analytical structures. Analysis on these and other groups is called harmonic analysis.

In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.

An understanding of group theory is also important in physics and chemistry and material science. In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories.
Physics examples: Standard Model, Gauge theory
Copied from wikipedia

Answer 5

A six pack of bud and a package of brats?

Answer 6

when people with the same problems meet as a group to discuss things with the help of a counsellor

Answer 7

I have tried to get into some groups for Bipolar, no luck, maybe because I'm Bipolar or something.

Answer 8

Please give us some more data to go on

Answer 9

do you mean
group therapy